Monday, April 28, 2014

A witch with a compass

Wilma the Witch was given a new electronic compass that could tell her which way was north. It pointed to true north rather than magnetic north.

So she looked at her compass, found out which direction was north-east, put the compass away in her pocket and set out on her broom in a north-easterly direction. She continued as straight as she could over the surface of the earth for 1,000 km, without adjusting left or right.

She then took out her compass and found that even though she had started out by heading north-east and had continued going straight she was no longer exactly on a north-east bearing.

So she now found out which way was east and headed east. This time, for a change, she keep looking at her compass and diligently maintained a constant bearing, ( i.e. she kept going exactly east ) and continued for 6,000 km.

Then she turned to the south-east and headed on a constant south-east bearing. After a while she found herself back at the starting point. She noted that she had crossed every line of longitude exactly once.

She decided to continue on a constant south-east bearing, she crossed the equator and eventually found herself at the south pole.

She then stopped.

How far did she travel in total?

You can assume that the earth is a sphere and that the distance from the equator to either pole it is 10,000 km.
Please also ignore any distance covered due to altitude gain traversing mountains.